By T. Mateos

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Non-Arbitrary Curve: If we now give up the restriction that the curve must be arbitrary, we can still show that the F1 and D0 projection 4 Note that it is not necessary that P commutes with M (y). 34 II. Physics of D-branes are necessary and sufficient, except for those cases in which the curve lies entirely in the flat directions that M8 may have. Of course, the former discussion shows that such projections are always sufficient, so we will now study in which cases they are necessary as well. In order to proceed, we need to prove an intermediate result.

It is then understandable that when any two branes are placed on top of each other the W -bosons become massless, and the gauge symmetry is enhanced from U(1) × U(1) → U(2). Putting all of the D-branes together just provides U(N) U(1) M= ∆Χ α ∆Χ us with a U(N) supermultiplet in (p + 1) dimensions with 16 supercharges, obtainable again from the ten dimensional one by dimensional reduction. Note that this includes a set of transverse scalar fields which, being in the same multiplet as the gauge fields, transform in the adjoint of the gauge group.

It is then clear that such projections will always commute with the F1 and the D0 ones, since they do not involve any gamma matrix of M8 . To complete the proof, one must take into account further possible problems that could be caused by the fact that the projections considered so far are applied to background spinors which are not necessarily constant. 36) with ǫ0 a constant spinor, and M(y i ) a matrix that involves only products of even number of gamma matrices on M8 (it may well happen that M(y) = ).